Olympiad problems

2001 Romania National Olympiad, 4

Tags: polynomial , algebra

Let $n\ge 2$ be an even integer and $a,b$ real numbers such that $b^n=3a+1$. Show that the polynomial $P(X)=(X^2+X+1)^n-X^n-a$ is divisible by $Q(X)=X^3+X^2+X+b$ if and only if $b=1$.

2014 Romania National Olympiad, 1

Tags: function , algebra

Find all continuous functions $ f:\mathbb{R}\longrightarrow\mathbb{R} $ that satisfy: $ \text{(i)}\text{id}+f $ is nondecreasing $ \text{(ii)} $ There is a natural number $ m $ such that $ \text{id}+f+f^2\cdots +f^m $ is nonincreasing. Here, $ \text{id} $ represents the identity function, and ^ denotes functional power.

2000 Tournament Of Towns, 2

Tags: geometry , parallelogram , isosceles

$ABCD$ is parallelogram, $M$ is the midpoint of side $CD$ and $H$ is the foot of the perpendicular from $B$ to line $AM$. Prove that $BCH$ is an isosceles triangle. (M Volchkevich)

1991 Baltic Way, 16

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Two circles $C_1$ and $C_2$ with radii $r_1$ and $r_2$ touch each other externally and both touch a line $l$. A circle $C_3$ with radius $r_3 < r_1, r_2$ is tangent to $l$ and externally to $C_1$ and $C_2$. Prove that \[\frac{1}{\sqrt{r_3}}=\frac{1}{\sqrt{r_2}}+\frac{1}{\sqrt{r_2}}.\]

2022 Israel National Olympiad, P2

Tags: algebra

Real nonzero numbers $a,b,c,d,e,f,k,m$ satisfy the equations \[\frac{a}{b}+\frac{c}{d}+\frac{e}{f}=k\] \[\frac{b}{c}+\frac{d}{e}+\frac{f}{a}=m\] \[ad=be=cf\] Express $\frac{a}{c}+\frac{c}{e}+\frac{e}{a}+\frac{b}{d}+\frac{d}{f}+\frac{f}{b}$ using $m$ and $k$.

1994 Romania TST for IMO, 2:

Tags: modular arithmetic , number theory

Let $ n$ be an odd positive integer. Prove that $((n-1)^n+1)^2$ divides $ n(n-1)^{(n-1)^n+1}+n$.

2012 Junior Balkan Team Selection Tests - Romania, 3

Tags: combinatorics , number theory

Positive integers $a, b, c$ have greatest common divisor $1$. The triplet $(a, b, c)$ may be altered into another triplet such that in each step one of the numbers in the actual triplet is increased or decreased by an integer multiple of another element of the triplet. Prove that the triplet $(1,0,0)$ can be obtained in at most $5$ steps.

2023 CCA Math Bonanza, I10

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Bryan Ai has the following 8 numbers written from left to right on a sheet of paper: $$\textbf{1 4 1 2 0 7 0 8}$$ Now in each of the 7 gaps between adjacent numbers, Bryan Ai wants to place one of `$+$', `$-$', or `$\times$' inside that gap. Now, Bryan Ai wonders, if he picks a random placement out of the $3^7$ possible placements, what's the expected value of the expression (order of operations apply)? [i]Individual #10[/i]

2004 Oral Moscow Geometry Olympiad, 3

Tags: equal angles , locus , geometry

Given a square $ABCD$. Find the locus of points $M$ such that $\angle AMB = \angle CMD$.

2006 China Second Round Olympiad, 10

Tags: geometry , 3d geometry , sphere

Suppose four solid iron balls are placed in a cylinder with the radius of 1 cm, such that every two of the four balls are tangent to each other, and the two balls in the lower layer are tangent to the cylinder base. Now put water into the cylinder. Find, in $\text{cm}^2$, the volume of water needed to submerge all the balls.

2016 Hanoi Open Mathematics Competitions, 12

Tags: geometry , fixed , circles

Let $A$ be a point inside the acute angle $xOy$. An arbitrary circle $\omega$ passes through $O, A$, intersecting $Ox$ and $Oy$ at the second intersection $B$ and $C$, respectively. Let $M$ be the midpoint of $BC$. Prove that $M$ is always on a fixed line (when $\omega$ changes, but always goes through $O$ and $A$).

1980 Swedish Mathematical Competition, 4

Tags: integration , analysis , inequalities

The functions $f$ and $g$ are positive and continuous. $f$ is increasing and $g$ is decreasing. Show that \[ \int\limits_0^1 f(x)g(x) dx \leq \int\limits_0^1 f(x)g(1-x) dx \]

2006 Bulgaria Team Selection Test, 3

Tags: combinatorics

[b] Problem 6.[/b] Let $m\geq 5$ and $n$ are given natural numbers, and $M$ is regular $2n+1$-gon. Find the number of the convex $m$-gons with vertices among the vertices of $M$, who have at least one acute angle. [i]Alexandar Ivanov[/i]

2024 CCA Math Bonanza, L2.1

Tags: function

Let $\tau(x)$ be the number of positive divisors of $x$ (including $1$ and $x$). Find \[\tau\left( \tau\left( \dots \tau\left(2024^{2024^{2024}}\right) \right)\right),\] where there are $4202^{4202^{4202}}$ $\tau$'s. [i]Lightning 2.1[/i]

2019-IMOC, N3

Tags: number theory

Prove that there exists $N\in\mathbb{N}$ so that for all integer $n > N$, one may find $2019$ pairwise co-prime positive integers with \[n=a_1+a_2+\cdots+a_{2019}\] and \[2019<a_1<a_2<\cdots<a_{2019}\]

2017 Switzerland - Final Round, 6

Tags: combinatorics

The SMO camp has at least four leaders. Any two leaders are either mutual friends or enemies. In every group of four leaders there is at least one who is with the three is friends with others. Is there always one leader who is friends with everyone else?

2020 Korea National Olympiad, 3

Tags: graph , combinatorics

There are n boys and m girls at Daehan Mathematical High School. Let $d(B)$ a number of girls who know Boy $B$ each other, and let $d(G)$ a number of boys who know Girl $G$ each other. Each girl knows at least one boy each other. Prove that there exist Boy $B$ and Girl $G$ who knows each other in condition that $\frac{d(B)}{d(G)}\ge\frac{m}{n}$.

2012 JBMO TST - Macedonia, 4

Tags: number theory

Find all primes $p$ and $q$ such that $(p+q)^p = (q-p)^{(2q-1)}$

2023 Indonesia TST, A

Tags: algebra

Let $a_1, a_2, a_3, a_4, a_5$ be non-negative real numbers satisfied \[\sum_{k = 1}^{5} a_k = 20 \ \ \ \ \text{and} \ \ \ \ \sum_{k=1}^{5} a_k^2 = 100\] Find the minimum and maximum of $\text{max} \{a_1, a_2, a_3, a_4, a_5\}$

2007 Today's Calculation Of Integral, 211

Tags: calculus , integration , parabola , hyperbola , geometry , calculus computations , conic

When the parabola which has the axis parallel to $y$ -axis and passes through the origin touch to the rectangular hyperbola $xy=1$ in the first quadrant moves, prove that the area of the figure sorrounded by the parabola and the $x$-axis is constant.

1967 IMO Longlists, 27

Tags: geometry , 3d geometry , polygon , cube , intersection

Which regular polygon can be obtained (and how) by cutting a cube with a plane ?

2015 VTRMC, Problem 6

Tags: functional equation , fe , absolute value

Let $(a_1,b_1),\ldots,(a_n,b_n)$ be $n$ points in $\mathbb R^2$ (where $\mathbb R$ denotes the real numbers), and let $\epsilon>0$ be a positive number. Can we find a real-valued function $f(x,y)$ that satisfies the following three conditions? 1. $f(0,0)=1$; 2. $f(x,y)\ne0$ for only finitely many $(x,y)\in\mathbb R^2$; 3. $\sum_{r=1}^n\left|f(x+a_r,y+b_r)-f(x,y)\right|<\epsilon$ for every $(x,y)\in\mathbb R^2$. Justify your answer.

2011 Tournament of Towns, 2

Tags: analytic geometry , geometry , combinatorics

Passing through the origin of the coordinate plane are $180$ lines, including the coordinate axes, which form $1$ degree angles with one another at the origin. Determine the sum of the x-coordinates of the points of intersection of these lines with the line $y = 100-x$

2009 Estonia Team Selection Test, 6

Tags: number theory , divisor , product

For any positive integer $n$, let $c(n)$ be the largest divisor of $n$ not greater than $\sqrt{n}$ and let $s(n)$ be the least integer $x$ such that $n < x$ and the product $nx$ is divisible by an integer $y$ where $n < y < x$. Prove that, for every $n$, $s(n) = (c(n) + 1) \cdot \left( \frac{n}{c(n)}+1\right)$

2013 NIMO Problems, 1

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What is the maximum possible score on this contest? Recall that on the NIMO 2013 Summer Contest, problems $1$, $2$, \dots, $15$ are worth $1$, $2$, \dots, $15$ points, respectively. [i]Proposed by Evan Chen[/i]